An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

Staffing a service system with appointment-based customer arrivals

Appointment systems are widely used to facilitate customers’ access to services in many industries such as healthcare. A number of studies have taken a queueing approach to analyse service systems and facilitate managerial decisions on staffing requirements by assuming independent and stationary customer arrivals. This paper is motivated by the observation that the queueing-based method shows relatively poor performance when customers arrive according to their appointment times. Because customer arrivals are dependent on their appointment times, this study, unlike queueing-based methods, conducts a detailed analysis of appointment-based customer arrivals instead of making steady-state assumptions. We develop a new model that captures the characteristics of appointment-based customer arrivals and computes the probability of transient system states. Through the use of this model, which relaxes stationary and independent assumptions, we propose a heuristic algorithm that determines staffing requirements with aims to minimizing staff-hours while satisfying a target service level. The simulation results show that the proposed method outperforms the queueing-based method.     

Stationary analysis of the shortest queue first service policy

We analyze the so-called shortest queue first (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time, the queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called “symmetric case,” with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$\begin{aligned} M(z) = q(z) \cdot M \circ h(z) + L(z) \end{aligned}$$ for unknown function M, where given functions \(q, L\) , and \(h\) are related to one branch of a cubic polynomial equation. We study the analyticity domain of function M and express it by a series expansion involving all iterates of function \(h\) . This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the head-of-line preemptive priority system, which is the key feature desired for the SQF discipline.     

Strategic customer behavior in a queueing system with a loss subsystem

In the spirit of Little’s law \(L=\lambda W\) and its extension \(H=\lambda G\) we use sample-path analysis to give a general conservation law. For queueing models the law relates the asymptotic average workload in the system to the conditional asymptotic average sojourn time and service times distribution function. This law generalizes previously obtained conservation laws for both single- and multi-server systems, and anticipating and non-anticipating scheduling disciplines. Applications to single- and multi-class queueing and other systems that illustrate the versatility of this law are given. In particular, we show that, for anticipative and non-anticipative scheduling rules, the unconditional delay in a queue is related to the covariance of service times and queueing delays.     

The waiting time distribution for a correlated queue with exponential interarrival and service times

We are concerned with the analysis of the waiting time distribution in an $M∕M∕1$ queue in which the interarrival time between the $n$th and the $(n+1)$th customers and the service time of the $n$th customer are correlated random variables with Downton’s bivariate exponential distribution. In this paper we show that the conditional waiting time distribution, given that the waiting time is positive, is exponential.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

Stability conditions for a multiserver queueing system with a regenerative input flow and simultaneous service of a customer by a random number of servers

Queueing Systems - We investigate the stability condition of a multiserver queueing system. Each customer needs simultaneously a random number of servers to complete the service. The times taken by...     

Limiting expectation time distribution for a critical load in a system with relative priority

A single-channel system with relative priority and a hyperexponential input stream is considered. The limiting distribution of the virtual expectation time for lowest-priority requests under conditions of a critical load is found.     

Optimal utilization of service facility for ak-out-of-n system with repair by extending service to external customers in a retrial queue

In this paper, we study ak-out-of-n system with single server who provides service to external customers also. The system consists of two parts: (i) a main queue consisting of customers (failed components of thek-out-of-n system) and (ii) a pool (of finite capacityM) of external customers together with an orbit for external customers who find the pool full. An external customer who finds the pool full on arrival, joins the orbit with probability γ and with probability 1- γ leaves the system forever. An orbital customer, who finds the pool full, at an epoch of repeated attempt, returns to orbit with probability δ (< 1) and with probability 1- δ leaves the system forever. We compute the steady state system size probability. Several performance measures are computed, numerical illustrations are provided.     

On a loss system with a given number of customers

A one-server loss system with Poisson arrival stream and deterministic service times is considered conditional on the number of customers who appeared up to a givenT. This condition implies that the arrival times form a sample of the uniform distribution on (0,T]. We derive several characteristics of interest, such as the blocking probability at any given timet (0,T], the probability that exactlyi of the customers in (0,T] are served and, as a generalization, the distribution of the number of served customers arriving in any subinterval of (0,T].     

Waiting time analysis for a tandem queue with correlated service

The effect on overall waiting time including service is investigated when there is positive correlation between the first and second stage service times in a two stage tandem system. The importance of understage waiting (or “buffer”) space is underlined. The analysis is an application of the authors' work reported in [1].     

Stationary distributions in a queueing system with vacation times and limited service

This paper deals with a modified M/G/1 queueing system with finite capacity and a walking server. Units waiting are served up to a limited number before the server takes a vacation time and later returns to the queue again. A computational method for the stationary queue length distribution is developed and illustrated with a numerical example. The model was motivated by similar channel access mechanisms in token-ring local area networks.     

Optimal regulation of the service rate for a queue with finite waiting space

The optimal control for the service rate of a single-server queue with limited waiting space is derived. Costs are associated with providing the service and with waiting. A comparison with the traditional steady-state result is made.     

Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation—i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.     

Transient behaviour of a double-channel Markovian queue with limited waiting space

For a double channel Markovian queue with finite waiting space the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities are obtained in a simple closed form which can be easily used to find the important parameters of the system.     

Heavy traffic limit for a tandem queue with identical service times

We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.     

Equilibrium customer strategies in a single server Markovian queue with setup times

We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.      

The distribution of the number of arrivals in a subinterval of a busy period of a single server queue

In the course of attempting to estimate the arrival rate of a single server queue using an active probing experiment, the authors found it necessary to derive the distribution of the number of arrivals between two probes under the conditions that the busy period of the queue lasts this long. In this paper we derive this distribution. The key building blocks in the derivation of the distribution are the classical ballot theorem and its generalized forms.     

Sojourn times in a processor sharing queue with service interruptions

We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the on-periods are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.